290 ANALYTICAL MECHANICS
may be obtained from equations (5) and (III), respectively in equation (3) of page 82. Thus
2 /dn2 . ,/dey
7j2 ._/ — \ 4- r 2 / —
v U/ U/
= ^2sin*0+~ (7)
p2 Tl
Eliminating sin2 6 between equations (5) and (7) we get
2 h2 , o ,v , 2h*
V2 _ (6- - 1) H -- -
e2p2 ^ epr
--£1(^-1)+ 2*, (8)
e2p2 r
where k = — = const.
ep
9 If /}2
Therefore v2 - — = ™ (e2 - 1) = const. - (9)
r 62p-
221. Conditions which Determine the Type of the Orbit. —
Suppose a gravitating body to be projected into the field of another gravitating body, which acts as the center of force; then the type of the orbit is determined by the initial conditions, that is, the magnitude and the direction of the velocity of projection and the distance of the particle from the center of force at the instant of projection. Substituting the initial values of v and r in equation (9) and rearranging we obtain the following expression for the eccentricity :
The character of the orbit is determined by the value of the factor in the parentheses of equation (10). When it vanishes e is one, therefore the orbit is a parabola ; when it is negative e is less than one, therefore the orbit is an ellipse; and when it is positive e is greater than one, therefore the orbit is a hyperbola. We have, therefore, the following criteria :